Recommended fasts include Mondays/Thursdays, the three white days each month, six days of Shawwal, and عرفات/Ashura days — avoid fasting on Eid and Tashreeq days.
Voluntary fasting after حج is an excellent way to continue building on the spiritual foundation established during the حج. The most significant post-حج fasting opportunity is the six days of Shawwal — نبی کریم (صلی اللہ علیہ وسلم) said: 'Whoever fasts Ramadan then follows it with six days of Shawwal, it is as if they fasted the entire year' (Muslim). Since حج occurs in Dhul Hijjah (after Shawwal), this applies if you fast the following Shawwal.
Year-round مستحب fasts include: Mondays and Thursdays (نبی کریم's regular practice, as these are days when deeds are presented to Allah), the three white days of each lunar month (13th, 14th, and 15th — Ayyam al-Beed), the Day of عرفات (9th Dhul Hijjah, for non-حجاج), and the Day of Ashura (10th Muharram, with the 9th or 11th). Important: fasting is prohibited on the two Eid days and the three days of Tashreeq (11th-13th Dhul Hijjah), so new حجاج should not attempt to fast immediately after their حج rituals conclude.
Source: Sahih Muslim; Sunan al-Tirmidhi; Imam al-Nawawi, Riyad al-Salihin